Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d

Gilles Lebeau

doi:10.1051/cocv:2002052

Abstract

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l'équation d'Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t=0 lorsque u est définie sur un demi-interval [O,T[.

References

G. Birkhoff, Helmholtz et Taylor instability. Proc. Symp. Appl. Math XIII. Amer. Math. Soc. (1962) 55-76.

Bardos, C., Frisch, U., Sulem, C. et Sulem, P.L., Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability. CMP 80 (1981) 485-516.

Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. Ec. Norm. Sup. IV 14 (1981) 209-246. CrossRef

Delort, J.-M., Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991) 553-586. CrossRef

Duchon, J. et Robert, R., Global vortex sheet solutions of Euler equation in the plane. J. Differential Equations 73 (1988) 215-224. CrossRef

G. Lebeau, Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2d. Séminaire X-EDP 2000/2001, exposé 1 (2000).

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Lebeau, Gilles 2002. Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d. ESAIM: Control, Optimisation and Calculus of Variations, Vol. 8, Issue. , p. 801.

Lopes Filho, Milton C. Lowengrub, John Nussenzveig Lopes, Helena J. and Zheng, Yuxi 2006. Numerical evidence of nonuniqueness in the evolution of vortex sheets. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 40, Issue. 2, p. 225.

Secchi, Paolo 2007. Analysis and Simulation of Fluid Dynamics. p. 201.

Ambrose, David M 2007. Regularization of the Kelvin–Helmholtz instability by surface tension. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 365, Issue. 1858, p. 2253.

Бардос, Клод Bardos, Claude Тити, Эдрис С and Titi, Edriss S 2007. Уравнения Эйлера идеальной несжимаемой жидкости. Успехи математических наук, Vol. 62, Issue. 3, p. 5.

Constantin, Peter 2007. On the Euler equations of incompressible fluids. Bulletin of the American Mathematical Society, Vol. 44, Issue. 4, p. 603.

Kamotski, V. and Lebeau, G. 2008. On 2D Rayleigh-Taylor instabilities. Journées équations aux dérivées partielles, p. 1.

Shatah, Jalal and Zeng, Chongchun 2008. A priori estimates for fluid interface problems. Communications on Pure and Applied Mathematics, Vol. 61, Issue. 6, p. 848.

Bardos, Claude Titi, Edriss S. and Linshiz, Jasmine S. 2010. Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations. Communications on Pure and Applied Mathematics, Vol. 63, Issue. 6, p. 697.

Benzoni-Gavage, Sylvie Coulombel, Jean-François and Tzvetkov, Nikolay 2011. Ill-posedness of nonlocal Burgers equations. Advances in Mathematics, Vol. 227, Issue. 6, p. 2220.

Castro, A. Córdoba, D. and Gancedo, F. 2012. Mathematical Aspects of Fluid Mechanics. p. 88.

Alazard, T. Burq, N. and Zuily, C. 2012. Low regularity Cauchy theory for the water-waves problem: canals and swimming pools. Journées équations aux dérivées partielles, p. 1.

Wang, Chao and Zhang, ZhiFei 2012. A new proof of Wu’s theorem on vortex sheets. Science China Mathematics, Vol. 55, Issue. 7, p. 1449.

Milgrom, Timur and Ambrose, David M. 2013. Temporal boundary value problems in interfacial fluid dynamics. Applicable Analysis, Vol. 92, Issue. 5, p. 922.

Sueur, Franck 2015. Viscous profiles of vortex patches. Journal of the Institute of Mathematics of Jussieu, Vol. 14, Issue. 1, p. 1.

Ambrose, David M. 2016. Lectures on the Theory of Water Waves. p. 140.

Ambrose, David M. 2016. Small strong solutions for time-dependent mean field games with local coupling. Comptes Rendus. Mathématique, Vol. 354, Issue. 6, p. 589.

Caflisch, R. E. Gargano, F. Sammartino, M. and Sciacca, V. 2017. Regularized Euler- $$\alpha $$ α motion of an infinite array of vortex sheets. Bollettino dell'Unione Matematica Italiana, Vol. 10, Issue. 1, p. 113.

Izosimov, Anton and Khesin, Boris 2018. Vortex sheets and diffeomorphism groupoids. Advances in Mathematics, Vol. 338, Issue. , p. 447.

Morisse, Baptiste 2018. On hyperbolicity and Gevrey well-posedness. Part two: Scalar or degenerate transitions. Journal of Differential Equations, Vol. 264, Issue. 8, p. 5221.

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Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d

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Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d

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